Introduction to Scientific Computing II Multigrid & Steepest Descent Dr. Miriam Mehl

Multigrid – Some Rules smoother optimal smoothing not!!! optimal convergence small number of smoothing iterations!

Multigrid – Some Rules grid coarsening standard: doubling of h exceptions: unisotropic operators adaptively refined grids unstructured grids/general SLEs

Multigrid – Some Rules restriction/interpolation order of restriction + order of interpolation > order of discretisation

Multigrid – Some Rules V-cycle faster W-cycle more robust

Steepest Descent – Basic Idea solution of SLE minimization iterative one-dimensional minima direction of steepest descent?

Steepest Descent – Principle

Steepest Descent – Principle

Steepest Descent – Algorithm

Steepest Descent – Algorithm II

Steepest Descent – Example initial error after 1 iteration after 10 iterations

Steepest Descent – Example 1/128 1/64 1/32 1/16 h 48,629 11,576 2,744 646 iterations